Mathematical Research Letters

Volume 25 (2018)

Number 2

Purity of critical cohomology and Kac’s conjecture

Pages: 469 – 488



Ben Davison (Department of Mathematics and Statistics, Glasgow University, Glasgow Scotland, United Kingdom)


We provide a new proof of the Kac positivity conjecture for an arbitrary quiver $Q$. The ingredients are the cohomological integrality theorem in Donaldson–Thomas theory, dimensional reduction, and an easy purity result. These facts imply the purity of the cohomological Donaldson–Thomas invariants for partially nilpotent representations of a quiver with potential $(\widetilde{Q},W)$ associated to $Q$, which in turn implies positivity of the Kac polynomials for $Q$.

Received 27 January 2014

Accepted 21 April 2017

Published 5 July 2018